orientation$−$ preserving or orientation$-$reversing in the Definition of Teichmuller space

There are several questions about the definition of Teichmuller space.

Definition(see A Primer on Mapping Class Groups)

1. Let $S$ be a genus g closed surface with $g \ge 2$.The Teichmuller space of $S$,$Teich(S_g)$, is defined to be homotopic classes of marked hyperbolic surfaces. Two marked hyperbolic surfaces $(X_1,\varphi_1)$,$(X_2,\varphi_2)$ are said to be homotopic if and only if there is an isometry $I:X_1 \to X_2$ such that $I \circ \varphi_1$ is homotopic to $\varphi_2$ in the usual sense.

2. Let $S=T^2$ ,The Teichmuller space of $S$,$Teich(T^2)$, is defined to be homotopic classes of marked unit-area flat surfaces on $T^2$.

Questions

In this definition, Are both the markings and the isometry required to be orientation$-$ preserving?

The followings are my understandings:

• First of all, since we finally defined a Teichmuller metric on this space, the change of marking map must be orientation$-$ preserving, and thus the isometry in the definition must be orientation$-$ preserving. the requirement of orientation$-$ preserving for the isometry can also be seen from the fact that we always identify hyperbolic surfaces with Riemann surfaces and the isometry with biholomorphism.

• Second, according to the definition for twist parameters $\theta_i$, markings should also be orientation$-$ preserving, otherwise, we can easily construct two marked hyperbolic surfaces with the same Fenchel-Nielsen Coordinates, which means the map $FN$ in the Theorem 10.6 is not injective.

All of those two points can actually be easily seen from the fact that the definition for twist parameters $\theta_i$ should be well$-$defined.

So we may suppose that those maps need to be orientation$-$ preserving. But the problem is that:

• In another model of Teichmuller space, namely, $DF(\pi_1(S_g),PSL(2,R))/PGL(2,R)$, can we replace $PGL(2,R)$ by $PSL(2,R)$? If we insist all of maps constructed in the proof of Proposition 10.2 to be orientation$-$ preserving, then every conjugation will come from $PSL(2,R)$,rather than in $PGL(2,R)$ . More concretely, if there is a conjugation come from orientation$-$ reversing, just as the reflection appeared in the step 2 of the proof of Proposition 10.1, it is difficult for me to understand the corresponding orientation$-$ preserving marking and isometry in Teichmuller space.

• Actually, this problem can also be explained in the definition of marked lattices,which required a choice of ordered generators ( ? $\blacktriangle$ But in the definition of high dimensional marked lattices in the page 354, we only need a choice of basis, and if we required an ordered basis, then $SL(2,R)$ acts $\clubsuit$ not transitively on the space of marked unit volume latices). $\bullet$ I don't know How to understand orientation$-$ reversing Euclidean isometries between marked lattices such as reflection about x$-$axis in the context of equivalence of marked hyperbolic surface under the assumptions that all maps are orientation$-$ preserving?( After all,in the second proof of Proposition 10.1, we only care orientation$-$ preserving linear map)

• Assuming that all maps appeared in the definition of Teichmuller space are orientation$-$ preserving, then How to define the actions of $Mod^{\pm}(S)$ on $Teich(S_g)$ by $f \cdot (X,\varphi)=(X,\varphi \circ f^{-1})$, since we now don't have orientation$-$ reversing marking.

• By Royden, $Mod(S)=Isom^{+}(Teich(S_g))$,but how to prove $Mod^{\pm}(S)=Isom(Teich(S_g))$?

• Generally don't use LaTeX for formatting. I also do not know what the triangle and club mean. Also I am guessing you want these "contradictions" to be addressed in an answer? – Paul Plummer Feb 27 '17 at 16:08
• I do not know about that book but traditionally, everything is oriented and orientation-preserving, including homotopy-equivalences. Thus, you work with $Mod^+$. Traditionally, you work with Riemann surfaces which are oriented by default; when working with hyperbolic surfaces instead, you should say "oriented hyperbolic surface". As for Royden's proof, that's a separate story. You can say that $Isom^-(T_g)$ contains an element given by an orientation-reversing self-homeomorphism. In view of this, everything reduces to the oriented case. – Moishe Kohan Feb 27 '17 at 17:28
• @MoisheCohen Since the way of understanding Teichmuller space via marking riemann surfaces is so natural for me, so I also hope I can explain the action of orientation-reversing map of Teichmuller space by the same formula as the preserving one, that is , f⋅(X,φ)=(X,φ∘f^{−1}). This cannot reduce to the oriented case – BiM Feb 28 '17 at 4:31
• @PaulPlummer This is my first time to use stackexchange and the triangle is indeed using for emphasizing such contradictions. – BiM Feb 28 '17 at 4:36